The last ingredient for our approach is selection of the distance metric \(\mathcal {D}(\cdot ,\cdot )\) . We used SWD for this purpose. SWD is an approximation to high dimensional Wasserstein Distance (WD) [1]}, [2]} which has been shown to be helpful in deep learning. Computing WD directly is challenging, however WD can be empirically approximated via SWD [3]}. SWD achieves this by approximating the high dimensional optimal transport problem via 1D Wasserstein Distance instances which are obtained by performing \(L\)  random unit sphere projections \(o_i\)  of its two input distribution as:
\(\mathcal {D}(P,Q) = \frac{1}{L} \sum _{i=1}^L \mathcal {D}_{1-WD} (\langle P, o_i \rangle , \langle Q, o_i \rangle ),\) 
